First list all the terms out.
e^ix = 1 + ix/1! + (ix)^2/2! + (ix)^3/3! ...
Then, we can expand them.
e^ix = 1 + ix/1! + i^2x^2/2! + i^3x^3/3!...
Then, we can use the rules of raising i to a power.
e^ix = 1 + ix - x^2/2! - ix^3/3!...
Then, we can sort all the real and imaginary terms.
e^ix = (1 - x^2/2!...) + i(x - x^3/3!...)
We can simplify this.
e^ix = cos x + i sin x
This is Euler's Formula.
What happens if we put in pi?
x = pi
e^i*pi = cos(pi) + i sin(pi)
cos(pi) = -1
i sin(pi) = 0
e^i*pi = -1 OR e^i*pi + 1 = 0
That is Euler's identity.
2 x 2 x 3 x 3
This is really easy to search up btw
In order to solve this problem, you must draw a right triangle first to help you visualize the path of the jogger.
You are given 0.5 miles north (which will be the height of the right triangle), and 1.3 miles (which will be the hypotenuse of the triangle).
Using Pythagorean Theorem: a²+b²=c², we have:
0.5² + b² = 1.3²
b = 1.3²-0.5²
b = 1.2 miles
Answer:
<em>The number of </em><em>pine tree is 119</em><em> and the number of </em><em>elm tree is 187.</em>
Step-by-step explanation:
The ratio of pine trees to elm trees in a park is 7:11
Let us assume that the number of pine tree is 7x and the number of elm tree is 11x.
It is also given that, there are 68 more elm trees as compared to the pine trees.
i.e 
Hence, the number of pine tree = 
and the number of elm tree = 
